The ∞‐Besov capacity problem

A theory of ∞‐Besov capacities is developed and several applications are provided. In particular, we solve an open problem in the theory of limits of the ∞‐Besov semi‐norms, we obtain new restriction‐extension inequalities, and we characterize the pointwise multipliers acting on the ∞‐Besov spaces.     

On the numerical approximation of p‐biharmonic and ∞‐biharmonic functions

The ∞ ‐Bilaplacian is a third‐order fully nonlinear PDE given by (1) In this work, we build a numerical method aimed at quantifying the nature of solutions to this problem, which we call ∞ ‐biharmonic functions. For fixed p we design a mixed finite element scheme for the prelimiting equation, the p‐Bilaplacian (2) We prove convergence of the numerical solution to the weak solution of and show that we are able to pass to the limit p → ∞ . We perform various tests aimed at understanding the nature of solutions of and we prove convergence of our discretization to an appropriate weak solution concept of this problem that of ‐solutions.     

Global C∞‐solutions to 1D compressible Navier–Stokes equations with density‐dependent viscosity

In this paper, we consider the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries. The initial density ρ₀∈W^1,2n is bounded below away from zero and the initial velocity u₀∈L²ⁿ. The viscosity coefficient µ is proportional to ρ^θ with 0<θ?1, where ρis the density. The existence and uniqueness of global solutions in Hⁱ([0,1])(i = 1,2,4) have been established in (J. Math. Phys. 2009; 50 :023101; Meth. Appl. Anal. 2005; 12 :239–252; J. Differ. Equations 2008; 245:3956–3973; Commun. Pure Appl. Anal. 2008; 7 :373–381). By mathematical induction method, we will establish the existence of global smooth solutions to 1D compressible isentropic Navier–Stokes equations with density‐dependent viscosity and free boundaries when the initial data ρ₀ and u₀ are smooth. Copyright © 2011 John Wiley & Sons, Ltd.     

Lorenz‐gauged vector potential formulations for the time‐harmonic eddy‐current problem with L∞‐regularity of material properties

In this paper, we consider some Lorenz‐gauged vector potential formulations of the eddy‐current problem for the time‐harmonic Maxwell equations with material properties having only L^∞‐regularity. We prove that there exists a unique solution of these problems, and we show the convergence of a suitable finite element approximation scheme. Moreover, we show that some previously proposed Lorenz‐gauged formulations are indeed formulations in terms of the modified magnetic vector potential, for which the electric scalar potential is vanishing. Copyright © 2007 John Wiley & Sons, Ltd.     

On the computation of the infimum in H∞‐optimization

In this paper, a new method for the computation of the infimum for a large class of continuous‐time H_∞ optimal control problem by state feedback is presented. The main ingredients of the new method include three generalized eigenvalue problems whose coefficient matrices are from a condensed form of the given system. This condensed form is computed using only orthogonal transformations which can be implemented via a numerically stable way. The superiority of the new method over the existing one given in Chen (H_∞ Control and its Applications, Chapter 5. Springer: Berlin, 1997) is verified by some numerical examples. Copyright © 2004 John Wiley & Sons, Ltd.     

A Lie symmetry analysis and explicit solutions of the two‐dimensional ∞‐Polylaplacian

In this work, we consider the Lie point symmetry analysis of a strongly nonlinear partial differential equation of third order, the ∞‐Polylaplacian, in two spatial dimensions. This equation is a higher order generalization of the ∞‐Laplacian, also known as Aronsson's equation, and arises as the analog of the Euler–Lagrange equations of a second‐order variational principle in L^∞. We obtain its full symmetry group, one‐dimensional Lie subalgebras and the corresponding symmetry reductions to ordinary differential equations. Finally, we use the Lie symmetries to construct new invariant ∞‐Polyharmonic functions.     

L∞‐solutions for the ultra‐slow reaction‐diffusion equation

The existence and uniqueness of solutions for a reaction‐diffusion ultra‐slow equation are proved. We also show that they can be extended up a maximal time and are stable as long as they exist. Symmetric and positive solutions are also proved to exist.     

Characterization of g∞,σ‐integral operators

The application of the general tensor norms theory of Defant and Floret to the ideal of (p, σ)‐absolutely continuous operators of Matter, 0 < σ < 1, 1 ≤ p < ∞ leads to the study of g_p′,σ‐nuclear and g_p′,σ‐integral operators. Characterizations of such operators has been obtained previously in the case p > 1. In this paper we characterize the g_∞,σ‐nuclear and g_∞,σ‐integral operators by the existence of factorizations of some special kinds. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Finite‐time l2 − l∞ synchronization for discrete‐time nonlinear chaotic systems via information‐constrained delayed feedback

This article is concerned with the problem of finite‐time synchronization control for a class of discrete‐time nonlinear chaotic systems under unreliable communication links. Our aim is to design a delayed feedback controller such that the resulting synchronization error system is stochastically finite‐time bounded with a guaranteed performance level over a finite time interval. Some sufficient conditions for the solvability of the above problem are established. A delayed feedback control scheme involving constrained information about the past state is presented. Finally, the Fold chaotic system is used to demonstrate the effectiveness of our proposed approach. © 2014 Wiley Periodicals, Inc. Complexity 21: 138–146, 2015     

Semi‐active ℋ∞ damping optimization by adaptive interpolation

In this work we consider the problem of semi‐active damping optimization of mechanical systems with fixed damper positions. Our goal is to compute a damping that is locally optimal with respect to the ${?}_{\infty }$‐norm of the transfer function from the exogenous inputs to the performance outputs. We make use of a new greedy method for computing the ${?}_{\infty }$‐norm of a transfer function based on rational interpolation. In this paper, this approach is adapted to parameter‐dependent transfer functions. The interpolation leads to parametric reduced‐order models that can be optimized more efficiently. At the optimizers we then take new interpolation points to refine the reduced‐order model and to obtain updated optimizers. In our numerical examples we show that this approach normally converges fast and thus can highly accelerate the optimization procedure. Another contribution of this work is heuristics for choosing initial interpolation points.     

Robust reliable L2 – L∞ control for continuous‐time systems with nonlinear actuator failures

This article examines the reliable L₂ – L_∞ control design problem for a class of continuous‐time linear systems subject to external disturbances and mixed actuator failures via input delay approach. Also, due to the occurrence of nonlinear circumstances in the control input, a more generalized and practical actuator fault model containing both linear and nonlinear terms is constructed to the addressed control system. Our attention is focused on the design of the robust state feedback reliable sampled‐data controller that guarantees the robust asymptotic stability of the resulting closed‐loop system with an L₂ – L_∞ prescribed performance level γ > 0, for all the possible actuator failure cases. For this purpose, by constructing an appropriate Lyapunov–Krasovskii functional (LKF) and utilizing few integral inequality techniques, some novel sufficient stabilization conditions in terms of linear matrix inequalities (LMIs) are established for the considered system. Moreover, the established stabilizability conditions pave the way for designing the robust reliable sampled‐data controller as the solution to a set of LMIs. Finally, as an example, a wheeled mobile robot trailer model is considered to illustrate the effectiveness of the proposed control design scheme. © 2016 Wiley Periodicals, Inc. Complexity 21: 309–319, 2016     

Generalization of the B *‐algebra (C0(X),∥ ∥∞)

We give a generalization of the Stone–Weierstrass property for subalgebras of C (X), with X a completely regular Hausdorff space. In particular, we study in this paper some subalgebras of C₀(X), with X a locally compact Hausdorff space, provided with weighted norm topology. By using the Stone–Weierstrass property, we then describe the ideal structure of these algebras. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Fault‐tolerant sampled‐data mixed ℋ∞ and passivity control of stochastic systems and its application

This article addresses the problem of fault‐tolerant sampled‐data mixed and passivity control for a class of stochastic system with actuator failures, where the plant is modeled as a continuous‐time one and the control inputs are implemented as discrete‐time signals. Sufficient conditions for the reliable sampled‐data mixed and passivity performance control law is established for the considered systems by constructing an appropriate Lyapunov–Krasovskii functional together with the Newton–Leibniz formula and free‐weighting matrix technique. More precisely, linear matrix inequality based sampled‐data methodology is employed to design the mixed and passivity formation controller to reject the impact of the formation changes being treated as disturbances. Simulation studies are performed based on the flight control model to verify the stability, performance, and effectiveness of the proposed design strategy. © 2015 Wiley Periodicals, Inc. Complexity 21: 420–429, 2016     

Computing the norm ∥A∥∞,1 is NP-hard

It is proved that computing the subordinate matrix norm ∥A∥∞1 is NP-hard, Even more, existence of a polynomial-time algorithm for computing this norm with relative accuracy less than 1/(4n²), where n is matrix size, implies P = NP.     

L∞‐estimates for the Vlasov–Poisson–Fokker–Planck equation

We consider the Vlasov–Poisson–Fokker–Planck equation in three dimensions as the backward Kolmogorov equation associated to a non‐linear diffusion process. In this way we derive new L^∞‐estimates on the spatial density which are uniform in the diffusion parameters. Copyright © 2000 John Wiley & Sons, Ltd.     

Robust stability and H∞ control for nonlinear discrete‐time switched systems with interval time‐varying delay

This paper considers the problems of the robust stability analysis and H_∞ controller synthesis for uncertain discrete‐time switched systems with interval time‐varying delay and nonlinear disturbances. Based on the system transformation and by introducing a switched Lyapunov‐Krasovskii functional, the novel sufficient conditions, which guarantee that the uncertain discrete‐time switched system is robust asymptotically stable are obtained in terms of linear matrix inequalities. Then, the robust H_∞ control synthesis via switched state feedback is studied for a class of discrete‐time switched systems with uncertainties and nonlinear disturbances. We designed a switched state feedback controller to stabilize asymptotically discrete‐time switched systems with interval time‐varying delay and H_∞ disturbance attenuation level based on matrix inequality conditions. Examples are provided to illustrate the advantage and effectiveness of the proposed method.